Translation Process Approximation of a General Non-Gaussian Stochastic Process

Abstract

ATLSS Engineering Research Center, Lehigh University, USA.E-mail: pab409@lehigh.eduThe theory of non-Gaussian translation processes developed by Grigoriu imposescertain conditions between the non-Gaussian Power Spectral Density Function(PSDF) and the non-Gaussian marginal Probability Density Function (PDF) of thenon-Gaussian process. When these two quantities are compatible, it is straight-forward to estimate the PSDF of the underlying Gaussian process using translationprocess theory (the PSDF of the underlying Gaussian process is needed for simulationpurposes). However, the most challenging case for applications arising from real-life problems is when the arbitrarily prescribed non-Gaussian PSDF and PDF areincompatible. In this case, the objective is to approximate the incompatible non-Gaussian PSDF with a compatible non-Gaussian PSDF that is as close as possibleto the incompatible PSDF. To accomplish this, a number of methodologies have beenproposed that involve either some iteration scheme on simulated sample functionsor some general optimization approach. Although some of these techniques producesatisfactory results, they can be time consuming because of their nature (especiallywhen the prescribed non-Gaussian PDF deviates significantly from the GaussianPDF). In this work, a new iterative methodology is proposed that establishes theunderlying Gaussian PSDF with high accuracy and truly superior computationalefficiency. The basic idea is to iteratively update the Gaussian PSDF using thedirectly computed (through translation process theory) non-Gaussian PSDF at eachiteration, rather than through expensive ensemble averaging of PSDFs computedfrom a large number of generated non-Gaussian sample functions. The proposediterative scheme has the advantage of being very simple and converges extremelyfast. Once the underlying Gaussian PSDF is established, simulation of non-Gaussiansample functions is straight-forward without any need for iterations. A number ofnumerical examples are provided demonstrating the capabilities of the methodology.Keywords: stochastic process, translation process, non-Gaussian process, simulation,Spectral Representation Method.